Classification of angular curvature measures and a proof of the angularity conjecture

Intrinsic representation curves

The purpose of this article is to find a family of curves parametrized by arc length and that depend on an angular function and an intrinsic fraction function, which is defined as the quotient between torsion and curvature. We find for this family of curves explicit formulas of curvature, torsion and geodetic curvature, in terms of the angular function and the intrinsic fraction function. Applications are found for the case of the general helices and slant helices.

Yau猜想渐近数量曲率积分界的一个反例

A Counterexample to Yau's Conjectured Asymptotic Scalar-Curvature Integral Bound

通过结合环面透镜、角裤和自适应混合块，构造了正Ricci曲率完备单端三维流形，其数量曲率积分发散，给出Yau猜想反例。

埃尔米特integral geometry中的对偶曲率测度

Dual curvature measures in hermitian integral geometry

复空间型上的局部运动学公式导出了对偶酉不变曲率测度空间$\mathrm{Curv}^{\mathrm{U}(n)*}$上的交换代数的结构。建立在最近的结果，从积分几何在复杂的空间型，我们描述了这种代数结构明确作为一个多项式代数。这是一种编码所有局部运动学公式的简短方法。然后，我们刻画了复杂的空间型上的不变估值离开不变的角曲率措施固定的空间。

The Positive Mass Theorem with Angular Momentum and Charge for Manifolds with Boundary

Motivated by the cosmic censorship conjecture in mathematical relativity, we establish the precise mass lower bound for an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and minimal surface boundary, in terms of angular momentum and charge. In particular this result does not require the restrictive assumptions of simple connectivity and completeness, which are undesirable from both a mathematical and physical perspective.

Perelman灵魂定理在奇异空间中的推广

An extension of Perelman's soul theorem for singular spaces

本文研究具有非负曲率的开完全度量空间。其中，我们建立了Perelman灵魂定理在可能奇异空间中的推广：“设X是一个完备的、非紧的、有限维的Alexandrov空间，具有非负曲率。设X在非空开子集上无边界且有正曲率。则X一定是一个可缩空间“。这个结果的证明使用了距离函数和Busemann函数在具有非负曲率的奇异空间上的性质的详细分析。我们将引入一类角过剩函数来度量奇异空间中凸超曲面的凸性和外曲率。我们还得到了一个新的比较，在非负弯曲的空间，这导致了我们的新的灵魂定理的证明所需的凸性估计。